We introduce two famous (or notorious) open problems from Smale's list and Millennium problems: the Jacobian conjecture and the Birch--Swinnerton-Dyer conjecture. 

The Jacobian conjecture states that if the Jacobian of a polynomial map is a nonzero constant, then the map is bijective. The condition of the Jacobian being equal to a constant can be translated to a system of (too many) polynomial equations. An elementary but promising approach is to find ALL solutions systematically. This is based on joint work with Jacob Gliedwell, William Hurst, Li Li, and George Nasr. 

 A special case of the weak BSD conjecture can be translated to counting integer solutions of certain Diophantine equations. Again an elementary but promising approach is to find ALL solutions systematically. This is based on joint work with Ty Clements, Zach Socha, and Vishak Vikranth.

This talk will be completely elementary and accessible to non-experts, as all my collaborators including myself are non-experts.